where we have used \(\sum _J \mathbb {T}_{Jk} = 1\)  to simplify the denominator. The above probabilities only depend on branching ratios, and therefore are invariant under time reparametrizations.This should be obvious, and to prove it we only need (REF ) to infer that \(\frac{\kappa _{Ij}}{\kappa _j} = \frac{\kappa ^{\text{proper}}_{Ij}}{\kappa ^{\text{proper}}_j}\) . They coincide with Bousso's “prior probabilities" [1]}, derived following a different line of reasoning, and are also closely related to Garriga and Vilenkin's “comoving" probabilities [2]}, [3]}.
In our approach, these probabilities are an inevitable consequence of justified objective reasoning with complete ignorance about the time of existence.
